I think the fact that you have an available closet behind your tank (if you are willing to go through the wall)
allows for you to have just about any type of diy filtration and you are not limited to the space under your tank. With all that space think big.
I think, in your case, with the closet being behind the tank, a DIY canister/sump combo would be a better bet than a tyical sump. You can make yourself a canister with 2 large sotrage totes or container (you can go pretty big, 40 gallons or so, so a 50% increase in the water capacity of your tank when accounting for filter media), corrugated hosing from a pool supply store, filter media, return pump.
Set the tote up with three or four chambers of sizes that you prefer custom fit to your media. Create chamber walls by cutting pieces of pvc
from one of the totes and using pvc
cement (approved for drinking water) to fuse the newly cut walls inside the "canister" tote. Leave a few inches of space from the top of the dividers for water to overflow into another chamber. Fill the first chamber with mechanical media (tons of pot scubbies works well here). Fill the second chamber with more of a fine material like poly-fill or a roll of batting. Leave another chamber for any chemical filtration you desire. Then the return chamber with the return pump where filtered water can collect and be pumped out of the canister.
Set it up just like you would a regular canister filter, with a siphon vacuum out the top of the tank, down the outflow hose about 12-18". You can then do a 90 turn to run straight through the wall into the closet and then another 90 turn straight down into the canister. Make sure the total fall of the water is 3x the length of the horizontal run through the wall.
Drill a bulkhead in the lid to the tote over the chaimber filled with coarse mechanical media. Let the hose run all the way through the bulkhead to within a couple inches of the bottom so the water has to fill up through the media and overflow into the next chaimber.
I think you get the idea and I'm sure there are some pretty important technicalities to this that need to be addressed. But if something like this interests you then I'm sure it can be set up to work right and well.