I have this old drill press and it has .004" play in the quill to the main casting play in the forward to column direction and .003" in the side to side direction.
I'll see that, and raise you three in both directions....
What would be an acceptable amount of play to have,
I don't think there's a "limit" to what's acceptable and what's not. You've got to remember that "drill press work" is not precise to begin with. If you're doing "router work" or "drum sander work", or other things that you "can" do with a drill press, you're gonna have the quill locked anyhow. For drilling, you can be out a whole bunch and it really won't bother much IF you're aware of it. It's like any other tool, it does the hard work and YOU do the brain work....
My drill press (Well, the one I keep at home. I've got a nicer one at work, due to bolting too the floor there 30 years ago when I didn't own a place to use it....), My drill press has a cracked base, a bent column, and a five o'clock Friday special head casting where the column and the quill bores are not quite parallel, and a non adjustable table that's not quite true to either. But I can still get a 3 inch deep hole in mild steel to come out the back within "standard clearance drill" dimensions, so stuff works out just fine. In my case, in that project, I just rotated the vise 180 degrees every half inch or so, so all the "crookedness" just kinda averages it's self out. Small pilot (I don't recall, but just a hair bigger than the web on the final drill is my SOP, and then a thirty second over 3/8 which is the bolt I was using (and still use) on those tool holders I was making....
That "turn the vise 180 thing is a great weapon against old drills, if you're going deeper than three or four diameters. For shorter than that, a simple center pop big enough to catch the drill point will bring it right on target. Of if it's a larger drill, start that hole with a smaller drill until THAT makes a big enough divot to catch the larger drill point. (or pilot the thing, you're already there...
That's just two things, there are SO many things you have to do to "squeeze" precision out of a brand new, tight, "perfect" drill press, that this isn't really any different, it's just that the "possible" errors are greater. As you figure out how that drill press "acts", you'll figure out what to do, and you'll find that three or four thousandths in the quill is NOT such a big deal.
In the case of having the quill loose in the casting bore, you're going to find another thing- If you "tease" the work with stupidly gentle drilling, you'll still probably get good results. After you're comfortable with what the drill press will do, remember that the rack and pinion gear that moves this is going to push that whole quill forward tremendously, "pinching" it against the front of the casting bore any time it's under anything more than a gentle axial load... In other words- Once you get done being scared of making a mistake, and just USE the drill, that particular wear kind of stops acting up (much)...
But... We're talking about a drill press, whose main job is to drill holes with drill bits, which don't make a hole on size anyhow, they're (probably) not as straight as you'd like to think they are anyhow, and generally, those themselves are not a precision tool. They'll wander off even with the best of precision machines, and just in general, the whole thing is not a precision operation.
I'm use to high quality milling machines and in this drill press it seems like a lot.
Drill presses tend to be not so precise, because frankly, they don't NEED to be that precise. They're less tight, less rigid, less accurate, and in the end, it sounds counterintuitive at first, but the lack of overall machine stiffness, the lack of precision, and honestly, the lack of a terribly large or terribly rigid quill, that's what allows YOU turn out good work from a lighter, faster, simpler machine.
At that level of wear, use the drill. If you're drilling two by fours, it's a drill, it'll work. If you're drilling metal parts, you'll quickly figure out this machine's personality. (They ALL have some "personality" if you're trying to squeeze precision out of 'em.)
And I might be wrong,I'm 98 percent sure that you have a "wedge" style of quill lock (no split casting), and so I would NOT recommend at all that you run it with the quill lock "snug", as that's gonna do more harm than good. (I still recommend it only sparingly on a split casting "pinch" style, but in that case, it's a lot less damaging to the machine....).
And that 3 and 4 thousandths you've measured- If you keep some light oil on that every now and again, you great, great grand kids are going to be able to take their pocket laser interferometric interpolator gauge and measure the same three or four thousandths of an inch...