Linoleum is easy to cut and is relatively cheap. You can do parts of a floor at a time; you don’t have to do the whole thing. I am not too sure that it absolutely has to be glued to the existing floor—if it is a small area, I think it will just lie there and do whatever it is you want it to do. If the landlord objects, you can just pick up the pieces and throw them out the windows, late at night.
If you do Poly: Use a water based Polyuerthane for floors. Fast drying, low fumes. Minwax has such a product in most home centers.
Can you do poly?
If you are willing to invest 75 bucks and several hours = yes.
If the floor generally sucks, you might be in 'who gives a $hit mode" and just start spreading the Poly. You might want to make some effort to sand bad areas, or use a stain to darken areas to get some consistency.
There are sanding blocks that hook up to broom handles, so you can walk around and do some light sanding.
Is Polly listed under massage or escort? 
What! You want cheaper than this???
Forty bucks’ll do most floors (9X12). And you can find cheaper, especially in floor covering stores. And messes clean up easy. Don’t lissen to all those other high-falootin’ ideas. For cheap, linoleum rocks! 
BTDT.
Less than perfect hardwood floors look better than great “linoliuem”.
Some aspects of this post verge on TMI!!
Poly
Landlord checking in. A deal I often make with my tenants in situations like this is that I will pay for the materials if they will do the labor. You might suggest that to your landlord. It’s a win-win solution.
Waxing is obviously not the answer. It’s thin and wears off quickly. If the floor has been waxed recently you’ll want to deal with it somehow. There’s a product called liquid sander which basically makes the surface tacky so that topcoats will hold. I’ve never tried it over wax, but it works fine over glossy paints.
Polyurathane is a pretty forgiving product and easy to apply. I recently had a professional flooring company redo the floors in this old house. After sanding and putting down a conditioner, they simply poured polyurathane directly from the can onto the floor and smeared it around with a big floor roller. That was it.
Of course, the ultimate solution may depend on how much you have invested - in an emotinal sense - in the place. Do you plan on living there a few more months or a few more years? Do you have any kind of budget in mind?
Is that how you spell it?
The OP’s floors, as described, are way beyond “less than perfect”, and as to which looks better, well, that’s an opinion. I’ve lived in rentals that were much improved by “sheet vinyl” flooring.
Other sins;
Tupperware
Aluminum pots from Kmart, etc
Penny’s jeans
Dinner at Mc D’s
Barber College haircuts
But I do read the OP.
My landlord is very cheap and very lazy; I am very lazy and very broke. I can spend $100 bucks or so; but I don’t want to pour too much money into a rental apartment.
Right now I have a sprained wrist, so this is going to have to wait awhile anyway.
Washable throw rugs are another possibility. I tried big area rugs and–well, yuck, after a few months with the cats.
Thanks for the water-based polyuerthane tip, I think I will buy some and try that.
Eve:
I’ve been a refinisher for years - of furniture and the occasional floor. You can work around the furniture if you like - up to you. Or you can move all the furniture to one side of the room, complete one side and then move it all to the other side.
Start with a clean floor - use wax remover first, or the poly won’t stick. Get 3 packages of sandpaper - one of 60 grit, one of 100 grit and one of 150 grit. Go around the area and Hand sand the areas that are the roughest or most splintery. You don’t need a sanding block if you don’t want to use one - just elbow grease. Start with the 60 grit and when the splinters are gone, go over it (until you like the smoothness - could only need a couple of passes or several) with the 100 grit, then give a few finishing strokes with the 150 grit (the finest). This will make sure that the splinters don’t start again. Vacuum well after sanding to get up all the dust, then mop LIGHTLY with pure water - use a clean (new, if possible) sponge mop and as little water as possible. This will get your surface as dust free as possible. Don’t use the same mop to apply the poly.
Then get water-based polyurethane (comes in gloss and satin (which is MUCH less shiny - almost dull) - your choice. Get a sponge mop with the most dense surface you can find - not the holey cellulous sponge kind, but one that has more of the texture of a foam paint brush - you can compare at the hardware store. ALso get a paint tray. You will probably need 3-4 cans of poly to do all the floor in a normal room.
Pour about an inch of poly in the paint tray and gently dip the surface of the mop in it coating it evenly - NEVER squeeze out the mop - this will make the poly foamy and difficult to work with. With gentle pressure (You are NOT mopping -this requires a different technique) Start in a corner of the room and bring the mop toward you with a gentle pressure and even stroke. Touch up a bit if you need to in one direction, but NO back and forth. Keep doing this throughout the room. Water-based poly dries very quickly - you can recoat in an hour, (no need to sand between coats. 3 coats are best to give a floor a long-wearing surface. If you missed any places, you can touch them up after the first coat is dry. An hour between coats and a few more hours drying time, and you’re good to go. My kitchen floor is wood, and I used water-based poly 10 years ago - it’s holding up fine.
PS - This should cost only about $50 - $75, including supplies.
Go rent a belt-sander, sand the crap out of your floor, then slather-on a finish-poly combo, whatever color you like. Two coats may be necessary. It’s not as cheap-ass lazy as some things, but dammit, there’s a wrong way and a right way to do cheap-ass lazy.
Did anyone else notice that Eve’s place is a rental? A cheap-assed rental owned by a landlord who’s too cheap-assed and lazy-assed to keep the apartment in habitable condition?
Me, if I had to, I’d buy the material and let the lazy-assed landlord do the labor. That is what you’re paying him for, after all. Make a deal, indeed.
Best bet; if at all possible do whatever it takes and buy a house. Then worry about quality and appearance. Renting really sucks, doesn’t it?
So, I’d put lino or something like it in the kitchen/bath and those rugs in the other areas. Paint the floors under the rugs with floor paint.
I’m gonna order a couple of these washable rugs for the two worst spots. Cats puke on 'em? Off to the laundromat.
I was gonna step in and say avoid the polyurethane. If it’s done badly (and if your underlying floor sucks), you’re likely to find the dried varnish peeling and cracking after a few weeks.
No kidding. I lived in a complete shithole apartment for a year. It was an old converted house, of which I had the first floor. The only reason they handn’t knocked it down already was that it was on some sort of historical register. I made a deal wherin I bought all the supplies and did all the work and deducted everything from rent. In the year I was there I don’t think I spent more than $20 on rent and got probably about a $800 in tools. Not to mention I had half the place into a really nice apartment. Unfortunately my roommate was not as up on the idea and I just slowy pushed back the GODAWFUL mess (we are talking junk piled chest height) that was his half.
Eventually I just got sick of having to live with half the apartment so nasty so I moved, but I made out well with tools and DIY experience. Ended up supporting myslef later as a handyman from what I learned.
Oh, and they knocked the place down inside of a year after I moved. All that work… pffft.
How the hell did this make “Threadspotting?” It’s probably the dullest thread on the whole Message Board this week!
Darlin’, a thread with you in it could never be dull!
Not to mention cat-puked rugs.
Well, I’m glad you found a solution. I was going to recommend teaching the cats not to throw up. For me, that would be preferable to refinishing a floor.