Can I swap out my ancient thermostat myself or is that a call-a-pro job?

I swapped out an old round mercury thermostat for a basic programmable one on my own boiler a few years back, and it was about a 30 minute job. No HVAC guy needed unless something surprises you.

Boilers are about the easiest thing you can hook a thermostat to. Your boiler is either a system like hot water baseboard or radiators, or it's a furnace. Either way, the thermostat is just a switch that says turn on. Most likely there are two wires back there, red and white, running at 24 volts. Some old installs even used three wires, and some of those old mercury ones had wires too short to reach a new stat's back plate, so take a photo of what's connected before you unscrew anything and note which wire went to which terminal.

Kill the power to the boiler at the switch or breaker before you touch the wires. At 24 volts it's not going to hurt you, but the boiler circuit sometimes has fused connections in the old junction box behind the stat, and tripping something isn't worth the hassle. Also, if your old one has a little glass vial full of silver liquid, that's mercury, and plenty of towns have a drop-off for it at the transfer station. Don't just toss it in the trash. On the new stat, buy one that runs on batteries, not one that needs a C wire, because with two wires you probably don't have one. Hardware store basic programmables will say right on the box if they're two-wire compatible for heat only. And if you see a little dial or slider on the back of either thermostat marked with numbers like 0.2, 0.3, called the heat anticipator, copy the setting from the old one onto the new one if you can read it, or just leave the new one at its default. Otherwise programming night setback is normal and boilers handle it fine. One heads up, don't recycle that old stat or leave it in a drawer and forget about it. Take one photo of the wiring, label the wires with masking tape, and it really is an afternoon job.

Did the same swap in my last place with cast iron radiators and one thing I learned: keep the night setback shallow, like 2 or 3 degrees max. A boiler with radiators takes hours to pull the house back up, so a big drop meant cold mornings while it caught up. Baseboard heats faster but same idea. Also mine had a third wire back there that turned out to be dead, just tucked it off to the side and it ran fine on two. Level the new stat on the wall too, a tilted digital one reads the room temp funny.

I'd push back on the 'most likely two wires at 24 volts' bit. Plenty of those ancient dial stats are line voltage, and some gravity-fed boiler setups ran on millivolts off the pilot. If the OP throws a modern 24 volt stat on one of those it either won't fire or cooks the internals. The easy check is to look at the old stat itself, they usally print the rating right on it. Mine said 120 volts and I had to hunt for a line voltage digital replacement, which took longer than the actual swap did. Also the heat anticipator talk is mostly moot, half the new stats don't even have one anymore.

Mine was a similar old dial setup and the only thing that made the job go smooth was a cheap multimeter I alredy had in a drawer. Put it across the two wires after killing power and it read right around 24 volts, so I knew a basic hardware store stat would work before I spent a dime. The other thing that saved me was the wires were so short they barely stuck out of the wall. I grabbed a couple of small wire nuts and pigtails to extend them, because otherwise the new back plate would not have reached. Took maybe an hour all in.