Can anyone tell me what they take the dirty laundry home in besides a paper bag? I just can't think of something washable.

Mesh bags with cord ties are sold in many places. Buy two, one to be filled and left there, one to carry home. Or even three.

https://www.amazon.com/Wanapure-Commercial-Available-Drawstring-Factories/dp/B07K69NHR2/ref=as_sl_pc_qf_sp_asin_til?tag=agin0c-20&linkCode=w00&linkId=c92080c6b80ce82c16a51307969f2ab1&creativeASIN=B07K69NHR2

Something like the Rubbermaid flex and carry is another option, it can't go in the washing machine but it's easy to clean and holds a lot
http://www.rubbermaid.com/en-US/flexn-carry%E2%84%A2-hamper

Hm, they appear to have discontinued it but there are lots of similar hampers available.

The dollar store has large laundry bags with a drawstring, or as Barb suggests, a king size pillowcase would work.

A pillowcase.

Also, there are also washable laundry bags that many folks use ( college students, folks who regularly haul laundry to a commercial laundry). Google "laundry bag".

This is a fairly vague question without alot of specifics but why not take it in one of the reusable bags that are very prevalent today for groceries. Many are made of fabric but almost all including plastic ones can go into the washer and hung to dry.