Gas / baseboard heating should be on but it's not -- how to investigate?
Based on the photos you provided I am sure that your heating system is hydronic, i.e. water is heated by the furnace and then flows through your baseboards giving up its heat to the room.  The unit on the left in your photo is your water heater which operates independently of your heating furnace (the unit on the right).  That's why you can have hot water without room heating. The large cylinder near the top of your furnace is the expansion chamber designed to handle the increase in water pressure as the water is heated by the furnace.  Your problem, however, may be due to the small unit to the right of the expansion chamber, the one with the electrical conduit attached. I believe that is a damper designed to close when the furnace is off to prevent heat being wasted by going up the flue.  When the furnace is on, that damper will be open to allow the gas combustion byproducts to go up the flue.  If that damper is not working properly, and stays closed, the furnace is designed to turn off to prevent those byproducts from filling your basement.  You should check to see if that is your problem.  If your thermostat is asking for heat, and that damper is closed, then the furnace will not run and you will not get heat. One way to cure this problem is simply to disconnect the damper as it does not increase the furnace efficiency very much anyway. Or you can have it replaced.  No guarantees, but this is easy enough to check. Another possibility is that the furnace pilot light is out or the thermocouple is defective.  You probably will need to call a repairman for these problems.