What is the verb meaning someone is stuck at home for such a long time that the person starts to exhibit psychological issues?

Possibly cabin as a verb which Lexico defines as:

Confine within narrow bounds

This would fit nicely as the same dictionary defines cabin fever as a North American informal term meaning:

Lassitude, irritability, and similar symptoms resulting from long confinement or isolation indoors during the winter.

Both terms come from the noun "cabin" which is a small dwelling often made from locally sourced timber or other material. "Cabin fever" seems to relate particularly to the psychological effects of being cooped up indoors by severe winter weather. The verb "to cabin" is not common but could well be what you are looking for.

The actual verb I think of is confine.

1 a : to hold within a location
       // Dikes confined the floodwaters.
1 b : IMPRISON

// the accused was confined until the trial could take place

Although psychological issues don't necessarily result, they sometimes can, especially if somebody, such as a prisoner, is mistreated during confinement.

During the COVID-19 lockdown, people have been confined to their homes.

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One expression that goes alone with confinement is stir-crazy:

slang
: distraught because of prolonged confinement

I'll propose cracked.

2 : mentally disturbed : crazy

It's an adjective, but in this use it derives from one of the senses of the verb crack.

3 : fail: such as

a: to lose control or effectiveness under pressure — often used with up.

// The stress was so great that she started to crack up.

I can't find a YouTube video related to looting as a side effect of the lockdown with "cracked" in the title, but it's in the right vicinity.

A possible use in such a title could be "People locked down for months have finally cracked."