Use Tide or Persil and BORAX with the hottest & longest cycle your washer has. Add extra rinsing. Body Oils are the culprit. Your water has to be very hot. 20 Mule Team Borax can be found along side the detergents at the supermarket.

No Tide or Persil for me -- Tide eats up my skin and I can't handle the scent of Persil, the strength of the fragrance rises up to the heavens BLECH. Never used borax or borax, might be worth a try. What is the difference between the two? I usually do two rinses, but like I said my washer sucks and "hot" isn't truly hot. Maybe I should boil a pot of water and dump it in there instead of just a pitcher-ful. Now I now next washer must have an on-board heater...

The problem may be the closet itself. Older items that were put away clean may have had traces of body oils on them that have gone rancid. It gives off a distinctive odor that travels to every item and puffs out into the room when the door is opened. You may have to empty the closet completely and re-wash everything in hot hot water to get ride of the odor. The closet itself would need to be cleaned and aired out also.

Try Biz powder as a detergent additive, and soak the sheets for an hour during the wash period. Biz has enzymes and such so may be a problem if you're sensitive. But bottom line is you need more-aggressive products and laundry habits to solve this problem.

I agree, sometimes the closet can have other smells in it that attach, sounds weird but I’ve definitely noticed it happening before. Try another location just to see.

Use a cup of vinegar in the rinse.

You never said what kind of washer you have. I have a GE compact front loader. Because is uses so little water, I usually do 5 rinses total for every load. I also have used BIZ. It's good for cutting oils too. BTW, there is a scentless Tide.

I have a Whirlpool top loader. I normally use deep water wash and 2 rinses with all loads.

Today when I put the sheets washed last weekend on the bed today (washed in powder, not liquid), the mustiness wasn't as bad -- still faintly there but not as bad as it was. Today I soaked the dirtied sheets using the powdered detergent, some vinegar, and poured boiling water in the machine, soaked for about 45 minutes or so, then ran through a wash cycle on hot. Will know how that worked when it's time to change the sheets next week.

(I didn't feel like dealing with the end-is-coming mayhem in the stores this morning just for some borax or ammonia, so I dumped in some vinegar instead).

I also forgot to say try Oxiclean White Revive liquid. I had some really nice sheets I found in the back of my closet that I wanted to keep but they had an odor and had yellowed. I started to throw them away but instead I soaked them overnight in Revive and followed up with a white vinegar rinse. They came out like brand new sheets. Love the stuff! Lol I cleaned my closet out, washed the walls down and went on a washing frenzy. :)

Vinegar may slightly address the odor but probably won't do anything useful to combat the source of it. Vinegar is mildly acidic which counters the alkalinity of detergent to balance the pH.

What cycle are you using on your Whirlpool? What is the model number?

The set you just washed, don't store them in the closet this time and see if you notice any difference.

Try soaking your sheets in DAWN dish washing liquid and hot water for a while, rinse them well, then launder them. DAWN is well known to be tough on grease.

Ah, I think I know the issue.

"I have a Whirlpool top loader. I normally use deep water wash and 2 rinses with all loads"

Seems as if you have one of the newer Whirlpool machines. Please let us know what the model number is so that we can help you out further more!

W10862396A

I'm getting new machines when I remodel the laundry/mud room, hopefully within the next few year (darn budget and all...) . I not-so-secretly wish the machines would blow up on me before then so I'd have a reason to replace them now.

I did put this week's sheets in the closet, that will let me know if the boiling water soak did anything to help. If not, I'll store out of the closet next wash.

"W10862396A"

That's not a valid Whirlpool model number, it's probably the document number from the user manual.

The model/serial tag is on the back of the loading port, behind/below the back edge of the lid.

I have the nose of a hound and can smell anything. I am wondering if your mattress has picked up odors-either from the sheets or from the room itself.

If you haven't yet, try the cup of ammonia that @dorey recommended above. It is the best at taking out grease and odors related to grease, imho. Just don't add chlorine bleach with it.