One thing to think about is the physical side of the job, lifting 200lb patients with your partner, carrying them down the stairs, etc. There are certainly people in their 50s with no problem with that, but a lot of people have accumulated back, knee injuries etc. Also worth thinking about hours, how would you feel about getting put on the overnight shifts? Third thing worth thinking about is pay. EMTs for private ambulance services are prob making something like $20-25,000 per year, medics around $40,000. So it's worth thinking about your savings, your financial comittments, etc.
Finally you should try to spend some time on a local ambulance if you can and see if you like it. Plenty of people who like medicine don't really like the stress of pre hospital work or dealing with really sick people in their own homes.
I think if you think about all those things and still want to do it age shouldn't really be a problem. Truth is that there is shortage of EMTs and medics in most parts of the country. And most EMTs stay in the field for three or four years from what I've seen, most medics for like 7. So companies aren't looking for people who are going to work for them for the next 20 years.
Also if you are thinking about working in EMS in a couple of years you might want to get started now. Many community colleges have EMT classes at night that you could take while still doing EMT. You could get your EMT, maybe do some volunteer EMT work etc so that when you are ready to do it full time you aren't starting from scratch.