It's absolutely possible, quite likely now. It would probably be too big a project to do anywhere but earth and maybe the moon right now. But the doors it would open if completed...

The Falcon series would be very limited for a moon mission. The Saturn V could get 47 metric tons into a trans lunar injection. Falcon 9 can get about 27 metric tons into GTO--not even to TLI (which isn't even listed in public information I could find, though one random Reddit post claims 3 metric tons). The Apollo lander was 17 metric tons, and it could take two people and a rover for a little tour on the surface. We can maybe shave some of that weight off with a new design, but probably not by half or anything really significant like that.

If we want to go back to the moon, it should be for more than taking pictures and picking up some rocks. You may not even be able to do that with a Falcon rocket.

NASA doesn't exactly rely on Starship for this, though. SLS does technically exist. It's just expensive, took far too long to build, and should probably be written off. Bezos might have something coming up, but who knows. Still relying on another space billionaire either way.

SpaceX had a brilliant track record for safety with their novel reusable rocket boosters. Even the first couple of Starship prototypes were incredibly successful, massively exceeding mission goals.

Unfortunately Musk seems to have entirely lost the sauce and is killing all of his companies, diving into conspiracy nonsense while funding an incredibly unpopular election campaign, gutting the federal government and tanking the economy by single-handedly raising the national unemployment rate through expensive and unnecessary layoffs. And during that same time Starship has become incredibly unreliable with prototypes not only failing to reach orbit but even exploding on the pad before attempting liftoff.

Meanwhile competitors are popping up around the world trying to recreate SpaceX's falcon rocket boosters, and many are starting to achieve success. Musk could have owned space but instead gestures wildly at everything and nothing in particular

Musk should have stepped down from all of his companies about 5-10 years ago and let them continue on without him. Maybe he'd run a funky tiny/manufactured home startup to try to "disrupt housing" or an online healthcare startup to try to "disrupt healthcare" or maybe he'd be running a drone startup to "disrupt warfare" or maybe he'd just sail off into the sunset impregnating as many women as he can convince to carry his kids while shitposting away on twitter. We can only dream only such an alternate reality