Hi and welcome to MA talk.
I am 35, well I will be in 3 days hee hee. Like you I started at 34 (just a couple of months ago) doing Aikido. I am 5'7" 240lb. completely out of shape and smoked till I started practicing...moreover never did any MA before.
Age and shape can be overcome. I still get exhausted and heavy breathing. Sometimes, when Sensei pushes hard, I reach the end of class that I can't even get up...actually Sensei sometimes calls the class off only when he sees that we are really on the limit. Getting used to work outs from a 0 activity takes some time, you might be still be getting heavy breath when other in the class are already getting used to the rythm.
The most important thing is start slowly. That doesn't mean you don't have to get tired, but start moving your body carefully, don't push your strectches to the limit, take time to know your body, what it can do and what it cannot do (for now, but later on it will).
If you didn't do much sport or any MA at all in the past, the main factor is your brain! Your mind will tell you many times "what am I doing? Just give up, it is not for me". Well it is not true, your mind wants you just go back to the lazy comy life on the couch.
Another factor, that I am still fighting to overcome (mine is rolling on my head), is fear. You will have to do movements that you haven't done since you were a teenager or many new ones. The idea of rolling on your head, falling in a certain way, stretching a limb in a certain way or other things are things kinda hard to "digest" by your body. Thus your body will use its natural way to tell you that there is danger and that is by creating fear...and you might not be aware of that, like for me it took quite some time to realize that my main problem in failing doing certain things was due to fear.
But I see it as everything belonging to MA training. It is a world worth exploring and when you get used to it you will enjoy it.