Indeed you could. When I was a kid I remember one of our relations being some sort of laundromat czar. Don’t know the extent of his business, but my childhood impression was that he was a pretty rich guy, and I always used to wonder just what volume of coins from the laundromat machines was necessary to constitute his wealth. 
There’s a local laundromat that advertises that they can handle horse blankets. Obviously, they cater towards the equine set generally, and the horse racing industry specifically, but they’ll do any kind of laundry. Their slogan? “We’ll keep your jockeys clean.”
Yeah. I’m thinking, “Fly, wash, and dry.” Well, at least it rhymes.
As for the OP, I owned cats for decades. Never had a problem washing their bedding, but you do have to take care to get the hair out of it first. You can do it by brushing the bedding, or what I used to do: just taking it out on the back deck and giving it a good shake. The fur will fly, literally, but anything left over should not cause a problem.
That was going to be my suggestion, but … SDMB being what it is … it’s already there.
Our home laundry machines are robust as heck, so I wash the dog blankets here, but then I do as above. I generally still need to vacuum (or wipe with a damp rag) out the drum of the washer afterward (our Labrador retriever has his PhDog in Shedding), and – of course – take a putty knife to the dryer’s lint screen (actually, though it ends up choked, it comes clean surprisingly easily).
And I generally don’t take the blankets outside for a pre-shake, either. I should. I probably could. It’s one of my innumerable moral failings that I don’t.
So, while I agree with @LSLGuy and others’ take (just do it), the “above and beyond” approach would be – IMHO – to spend a couple shekels and do the “guilt-free” cycle after
Thank you, one and all!
Cat hair is probably the least offensive thing that will go into those washers today.
The washing machines in my apartment building have a setting for “Extra Rinse.” I’m pretty sure that’s designed for people who are washing diapers and other similarly “soiled” clothing.
Well, I know I’m a million times as humble as thou art! 
Twice a year my gf has me take her horse pads and blankets to a laundromat and launder them. I never really thought about it, I’ve seen worse things being laundered.
Band name.
As for brushing a blanket, I’ve found that the best way is to lay the blanket flat out on the floor and use a household vacuum cleaner. You’ll need a partner to stand on one edge of the blanket to keep it in place while you vacuum, and then you have to always orient the vac so that the motorized brushes only ever pull the blanket away from your partner (this means keeping the front of the vac assumed at your partner’s feet). This keeps the blanket taut and flat on the floor; any other configuration results in the blanket getting pulled in and wrapped around the brushes.
Using this method, you can brush the blanket more aggressively, extracting a lot more hair than you can with hand-brushing.
True. But then you have to clean out the guts of your hair-clogged vacuum. 
Take it to the Laundromat.
I always use a little bleach when I wash the dogs things. The hair is not really immense and I clean the filters but I like a little disinfectant.
I actually bought a carpet rake and I now rake before vacuuming because I got so tired of having to take the vacuum cleaner apart to clean the brush every other time I vacuum. Embarrassingly, the cat hair is not nearly as bad as all the human hair, most of it in my color.
We have way too much long human hair around here. Wraps around vacuum rollers. Gets in shower drains. Hairbrushes are always full. You’re liable to see it anywhere.
I’ve pulled a few off Bayliss’ head. I know it’s mine, 'cause, yes, I hug him regularly.
Hair is a problem.
Pet blankets and towels and other items are easier items to dehair.
Well you are bigger than the cat.
This reminds me of a related question I’ve had for a while: is there any harm (to cleanliness) if I washed rags used to clean (thus encrusted with non-flammable cleaning solutions and maybe shoe polish in addition to the dust and dirt) with my wearable clothes?
I don’t know if there’s an objective answer to that, but I instinctively shudder at the thought. I personally wash cleaning rags separately in hot water on the “power wash” setting, which would destroy any of my clothes, so I’ve never tried it. Rags with shoe polish on them especially seem like they could stain your clothes.
I wouldn’t recommend it.
thx for the morning smirk …
you might wanna take that phrase over to the Epitaph-treat
LSL is the guy who goes “all-in” …
Because of this thread, I asked my gf yesterday if her horse blankets were okay. Turns out they are, but since I asked she sent me to the laundromat with a huge basket of rags to be washed.
The laundromat was packed, with primarily unusual people, as usual. There was only one huge washing machine open, so I put my rags in it. I got change (16 quarters!) and started feeding them in, but after taking six of my quarters it refused any further interaction with me.
Fück it, I wasted enough time. So I got my rags situated and left. Some crone who was watching me yelled, “hey, you should mark that machine out of order”. I laughed and said, “yeah, I’ll get right on that”. She freaked out!! and screamed, “DON’T YOU SASS ME!”
And that’s why I hate my rare visits to the laundromat.
Shoe polish is wax. Nothing good happens when that gets hot. Why would you want to wash them anyhow? IME a polish-impregnated rag gives the best shine.
Anything else non-flammable (or fully evaporated if flammable) is fine, but I prefer to run rags separately from clothing; if nothing else, rags often shed a lot of fiber and I’d rather not be picking scads of threads off clothes later.