Can I build muscles with push-ups and sit-ups?
Yes, but don't limit yourself to just two exercises. What you should be looking up is body-weight exercises, or calisthenics.
You can absolutely build up a base of strength and supporting structure around your joints, especially your knees and shoulders, doing body-weight exercises. You will not get huge, and you will not build strength past a certain point - once you can do 10–20 reps of any exercise you're building muscular endurance, not strength. There is nothing wrong with that; just be realistic about what you're going to achieve and what you're not.
Go online and find a good variety of exercises so you're balancing out - push-ups and sit-ups only strengthen one specific movement each. Think about the rest of you and make sure you work legs and back, as well as the muscles that connect your upper and lower half. These connections are your core. Abs are just one part of your core.
Make sure you learn good form. YouTube is great for this. Then, when you do go to the gym, everyone who knows what they're doing will see you and say to themselves not "haha skinny guy" but "guy who wants to change and has taken the trouble to learn how to do it properly." Anyone whose opinion matters will respect that.