Anything can be done if you're willing to spend the money to do it. Whether it's cost effective or not is another matter.
Without seeing your situation, I'm going to guess that a replacement engine with electric start might be the most viable option.
But there comes a time when maybe the best thing to do is transfer the mowing and upkeep of the mowing machines to someone else? Just something to consider.
If you think going the electric start route is your best option, please post engine make and model information. A few pictures wouldn't hurt, either.