Here’s the inside

Replace with a standard round recessed flood. Gonna need drywall work.

You would need to add something to mount the base to, but something like this.

Without seeing the whole closet ...size?...other lighting?...etc. it is hard to be sure what else would work...in our closet we have a decorative fixture that hangs down a little ..with several bulbs...so the light shines down and radiates outward..it lights our 6x12 closet with no other recessed lights

Those old 1950s recessed lights are a pain to replace. I haven't found any modern equivalents.

It's hard to tell the size from your photo. The ones I know and loathe are 8" x 12" and that Home Depot luminaire that Handy linked to wouldn't cover the hole. If my math is right, a similar one 16" in diameter would.

If yours is an 8" x 8", the 12" light should cover the hole OK.

You'd have to mount a standard round junction box for it, centered in the hole. The problem you're likely to run into is that unless you're lucky and the installer left some slack, the cable connecting to the existing luminaire probably won't be long enough to reach a box in the center of the hole. Flying splices (no box) inside the ceiling are a hazard and a code violation.

You might have to find an accessible location where you can install a box for the splice to extend the cable. If there's an accessible attic above this, you should be OK. Otherwise you might end up having to fish a new cable run from the switch box, which wouldn't be a lot of fun.

No worry Sophie... no exposed bulbs...

I've found that the type of CFL bulb you have takes a long time to warm up, and even at peak output, they seem dim.

We recently replaced similar bulbs in our closets with LED bulbs, and the difference is impressive.

I would try a different bulb before replacing the whole fixture.

Anne’s idea was the first thing that came to mind to me....and would be the easy solution....but be sure to get the right kind for your fixture...it is enclosed so be sure there will be no heat issue ...also the fixture may have a limitation to how many watts..as it looks quite old replacing it may be the best option...I might also explore fluorescent fixtures.... but they may distort color a little if that bothers you...the positioning of the fixture seems odd as well... are there other fixtures in the closet?

I would replace with a LED thin panel light. Lots of different sizes available.

I have the fluorescent fixtures in my closets...I swapped the old bulbs for the new fancy LED bulbs. Sweet lord they are BRIGHT!! If you’re looking for lots of light and not pretty, it’s a good, cheap option.

Leds are a better way to go. Check out our lighting guide. I think you will find it of use. Good luck. iTrust Home Services Lighting Guide