What should I do about this smoke detector that beeps constantly?
Doesn't look like it has a battery.  How old is it? There should be a manufacture or expiry date written on the cover.

Many older AC powered detectors didn't have backup batteries.  That cable you unplugged is the AC power. If it's over 10 years old (or maybe it's 5 years now) it should be replaced anyways.

Replace the smoke detector. They aren't expensive. If it is acting up you don't want to depend on it when there is a fire.