Honeywell aquastat L8124L install
the aquastat is a brand name of water fired boiler controls.  As such it controls a [water] boiler... as opposed to a furnace which only heats air.

the control relies on a well or aquastat well which is a brass/copper thingamajig that is part of the temperature sensor which protrudes into the boiler to measure water temperature.  These get corroded and scaled up, insulating it from the hot water, causing what seems to be a problematic control unit (aquastat) and erratic behavior.

so to replace just the control unit and not the well, turn off electrical power to boiler and it's controls and replace just the electrical control unit.  It's mostly a DIY if your ok with working with 120VAC.  But there's a good chance any new control will act erractic because...

the problem is with the well, and the well needs to be replaced.  To replace the well because it protrudes into the water section of the boiler you will be shutting off any automatic fill then dealing with what water drains out of the boiler above the well fitting, then having to refill and purge out air.  After removing the well inspect it and determine if the boiler and hydronic system should be flushed with a cleaner.

you don't need an exact aquastat [boiler control] replacement.  There are better controls less expensive than that like a hydrostat 3250plus.

on any old boiler it's not smart to not replace the well if replacing the aquastat