Does my Apple Watch have to be at a certain distance from my phone to work?

Yes,I never take my phone when I exercise,after you've finished your workout and return home your watch uploads all the workout data to the phone through your paired bluetooth connection.

While your running the watch uses it's inbuilt GPS and accelerometer to record your distance,heart rate,speed etc.

Read the following webpage

https://support.apple.com/en-us/HT205547

A GPS only Apple Watch can still provide some functions away from the iPhone, just not ones that require access to the internet. For example it can play music, podcasts, audiobooks stored on the Apple Watch, it can track your exercise, it can tell you the time.

A GPS only Apple Watch away from its iPhone on a 5 mile walk, cannot make or receive phone calls, nor text messages. It cannot tell the weather. It cannot stream music, it cannot find your friends, and other things that would require using he Internet.

A Cellular Apple Watch can do those things without your iPhone being nearby. The Cellular Apple Watch will sharing your iPhone's phone number, and for this service you pay extra for the watch itself, and to activate the cellular service you pay your iPhone carrier a monthly fee. In the U.S. the fee is $10 per month added to your iPhone bill ($120/year). Because of the iPhone phone number sharing, the iPhone carrier must support the Apple Watch. So if you decided to get a Cellular model, make sure you carrier supports the Apple Watch

iPhone Cellular Carriers that support the Apple Watch, which models, the radio frequencies each model uses, and the country where they offer service:

https://www.apple.com/watch/cellular

Your carrier Must be listed by name in the above link. NVMO's that piggy back on another carriers network, unless explicitly mentioned, do not support the Apple Watch.