I’m not the OP, but…

I guess it depends on what the “generation ship” is doing, how many people are actually aboard, and how much cash we want to throw at it.

Just hanging around our solar system in a graveyard orbit with a fairly small generational crew? Probably doable with current tech if money was no object. You don’t even have to do much recycling if you don’t care about the cost. A human only needs 700-800kg of oxygen per year. Water can be recycled, but a human “only” needs 30,000 gallons for a lifetime. That’s 250,000 pounds of water… but you could launch that with a single ares V. Food is a challenge (between 70,000 and 80,000 pounds needed for a lifetime), but you can probably put enough raw nutrients up there to make an edible slurry drink that would stay viable for a few centuries… and you’d have plenty of light and water for hydroponic greenhouses for additional variety. Energy isn’t that hard, either. If we didn’t care about the obvious safety risk, we could launch enough nuclear fuel to run a ship for centuries using current tech, with plenty of spare reactors and fuel shot up there just in case. Use the water in the hull as rad shielding and micro meteorite protection (ice makes a decent shell that can be “repaired” with a little heat and a squirt gun). Build a nice sized space station for everyone to live in, put plenty of spare parts up there, and you’re golden.

You can reduce those needs significantly if you do even a bit of work trying to recycle and reuse water/air and focus on things like lab grown foods and growing spaces.

None of that is impossible with current tech. You could lob enough food, air, and water to keep a small crew alive for generations… if you wanted to… and had an insane amount of money to spend.

Going to another star? Forget about it.