iPhone 5s wont make or receive calls

1.

Remove and reinsert the SIM card.

If it doesn't help visit your carrier, maybe the SIM card is damaged, ask for a replaced (new) card. Normally it is free.

Ask also to check if the SIM tray is bent, as even a tiny misalignment can cause the SIM card to not fully connect with the internal contacts in your iPhone.

2.

See these Apple articles below:

If you can't make or receive calls on your iPhone

If you can't make or receive calls on your iPhone - Apple Support

If you can't send or receive messages on your iPhone, iPad, or iPod touch

If you can't send or receive messages on your iPhone or iPad - Apple Support

3.

iPhone 5s is > 9 years old, which also affects the battery life. The hardware remains old, so does the battery.

The chance is very small that you can still have the battery replaced.

If it is still possible, have the battery replaced.

The battery can be replaced by Apple or an Apple Authorized Service Provider.

See this Apple article: iPhone Battery Repair & Replacement - Apple Support

If you are going to replace the battery, see -> Get your iPhone, iPad, or iPod touch ready for service

Get your iPhone, iPad, or iPod touch ready for service - Apple Support

The price is approx $ 69.

If you want to know exactly, call Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

or 

call 1-800-MY-APPLE

4.

Read this before you buy a newer type iPhone.

What country are you in and who is your cellular carrier?

That device is no longer usable as a phone in the US. All US carriers require devices capable of Voice over LTE, which means an iPhone 6 or newer.