Professional pilot, but not aero engineer, here …
Using jet engines for direct lift, ie thrust aimed to offset gravity, is stunningly inefficient. An AV-8B Harrier can do it for only a few minutes before exhausting it’s fuel supply.
As a general rule of design, the larger the craft, the relatively more fuel it can carry; that’s why trans-oceanic aircraft are so big. Yes, it takes more gas to move the larger airplane, but as the machine gets bigger, the fuel carrying capacity (& hence range & endurance) go up faster than the marginal increase in fuel burn to carry the larger aircraft and fuel.
So any very small personal-sized vehicle is going to be hard-pressed to carry more than a few minutes of fuel tops.
The next problem is control. Imagaine you built a platform, or some thing that looked more or less like a motorcycle or jet-ski (PWC). Now imagine it balanced on a narrow pole stuck up it’s underside. Not bolted or welded to the end of the pole; just balanced.
Now climb up there & sit on it without falling off the pole. Meanwhile, I’ll put a big fan in the room and blast 10 or 20 or 60 mph wind at you from the front or sides or back, moving my fan at random.
Tough problem for you to stay balanced up there. That’s what the control system needs to solve. With modern computers, gyros, etc, the design is simple enough for the professional engineers in the aerospace industry, but the DIY homebrew crowd is gonna fnd it a bit tough.
And what will you use to provide the power to steer / balance with? You can use vectored thrust, where the engines pivot to push left/right & fore/aft as needed, but you lose maybe 10% of yuor power doing that, exacerbating the challenge of getting enough power & range into the thing.
My bottom line:
Can it be done? Sure.
Will it be pretty lame in performance & endurance: Yes.
Would it be totally cool to have one: You bet.