This cactus was doing great (it used to be round and "inflated", now it's collapsed), but suddenly it got sick. Is it salvageable?

The soil looks like regular generic 'house-plant' mix.
The majority of cacti use a cactus mix or add course sand.
Roots cannot remain moist. Chances are it held too much moisture.
The reaction is similar to too little water.
Beware of root-rot, if it's not too late!
There must be drainage. It appears there is no saucer under the planter which may indicate no drainage.
If so, couple that with the dense soil and there may be the problem.
See one of the links for plant care...
I'm guessing this is a:
* GYMNOCALYCIUM
http://www.plantcare.com/encyclopedia/spider-cactus-536.aspx
OR
*Stenocactus multicostatus
http://www.plantcare.com/encyclopedia/brain-cactus-983.aspx
others:
Coryphantha poselgeriana vallida
http://www.plantcare.com/encyclopedia/coryphantha-poselgeriana-vallida-1821.aspx

Lobivia jajoiana
http://www.plantcare.com/encyclopedia/lobivia-jajoiana-642.aspx

Notocactus uebelmanianus
http://www.plantcare.com/encyclopedia/notocactus-uebelmanianus-1798.aspx

Thelocactus setispinus
http://www.plantcare.com/encyclopedia/thelocactus-setispinus-545.aspx

The answers to ID questions I've seen here are on this site at: "Plant Encyclopedia Database".
The 'Search by Attributes' section is a fair source for Plant ID.
To help 'drill-down' selections or suggestions, it could use fields for: leaf shape/size, plant shape/size, color, to name a few.