Depending on the type of carburetor the engine has on it, it could be a bad diaphram. Remove, dismantle and clean the carburetor and replace the gaskets, seals and diaphrams. Just be careful to lay out all of the gaskets, seals and diaphram (diaphrams if more than one) and replace only what was removed. Many Carburetor Rebuild Kits come with gaskets, seals and diaphrams for many carburetors as some use the same gaskets, seals or diaphrams as others do while other pieces may be different and not compatible for the carburetor on your' engine. A friend of mine took his' weedeater to a Repair Shop and they put every gasket, seal and diaphram that came in the Rebuild Kit inside the carburetor on his' weedeater. They had it for over a year and could not get it to run. He got it back and gave it to me, saying that if I could fix it, then I could have it. I took it to my' workshop, removed and dismantled the carburetor and saw that many of the gaskets, seals and diaphrams did not match the ports on the carburetor. I took each one, one at a time matching them to the ports and setting the non-matching ones to the side. I re-assembled the carburetor with the gaskets, seals and diaphrams that matched the ports, re-installed it on the weedeater and it started and ran on the 3rd pull. I only spent about 45 minutes on the job and when done, I took the weedeater back to my friend and showed him the pile of about 10 to 12 extra gaskets, seals and diaphrams that the shop's employees had put into the carburetor which did not match.
I would recommend that in the future, that if you do not run the engine dry on gas before storing, then pour the gas out or drain it before you try to start the engine. Put one to two ounces of Fuel Stabilizer in the gas tank and about a 1/2 tank of fresh fuel. Let it sit for 10 to 15 minutes, then with the shutoff in the "off" position or if the engine does not have this, remove the spark plug wire, then pull the start rope a few times to get some fresh gas and fuel stabilizer into the carburetor (the fuel stabilizer will dissolve the gunk buildup from the old gas and actually condition the diaphrams, gaskets and seals). Let sit for 10 to 15 minutes again, then with engine set in the "on" or "run" position or spark plug wire re-installed, choke the carburetor and the engine should start within a few tries. My' Trash Pump with a Honda engine set for 3 years with old gas in the gas tank. I did what I wrote above and it started on the 2nd pull and never missed a lick all of the time I used it last year.
Good luck!
Frank