You can't use pushups alone as one of bodyweight exercises and ask if it will make you bigger or better body shape. Even for 1000 reps, doesn't matter.
There are also other exercises like Pullups, ABS, Dips..etc(several training styles for each) that you don't need a gym to perform and all combined if you part it for each day training, some 4-5 days per week, will give you a respectful body shape but it will mostly help in muscle cuts, strength, power than getting you bigger.
If you want a nice body shape(cuts), not necessary big then YES.

If you want to grow bigger, then you must start to search for alternatives and you'll mainly need a partner, like carrying on your back some heavy sand bag, carrying with your legs some weight for pullups, for Ab exercises too and so on...