Achieving this with out-of-the-box features from Shopify will be very difficult.
Of course, you can do it by implementing your own private app for your store only. Or you could look around Shopify app store for booking apps that meet your needs. I am sorry I cannot recommend any single one in particular as I haven't tried any.
Hope this helps!
You are most definitely able to do that. With Shopify, it isn't always about your customers purchasing a tangible object.
There are two ways that you can go about this. The first way would be where your customers purchase without booking a time. They would just buy the product that you are selling (in your case an appointment of some sort) and then they could call you at a later date to book their timeslot.
If you were to go this way you would add the service the same as you would any other product: Product > Add Product in your Store Admin.
Then you would just create the product (aka appointment) and make sure that you un-select "This is a physical product". That way it wouldn't ask for a shipping address from your customers.
When you are done you hit Save and your appointment product will be available for your customers to purchase.
If you are actually wanting to schedule specific times, though, then you will need to get an app. By the sound what you are looking to do an app would be the easiest and most streamlined option available. The best one that I can recommend for you is called Book that App. It allows your customers to schedule appointments at any time. There are also reminders so you will never miss an appointment. There are a few more apps in the app store, but this one is definitely high on the list in my opinion.
I hope this was able to answer your question! If you are wondering about anything else or have any further questions, please do let me know!