Rose here from Shopify. Thanks for reaching out!
First off, you can certainly fulfill orders that were processed and paid before putting your store on pause.
The only way to process payments while your store is paused is by manually inputting the payment method through draft orders. An example of this would be to create a draft order and take the payment over the phone. We have a guide to creating draft orders here.
If you are looking to move forward with pausing your store, you can find our step-by-step guide here.
I hope that helps! If you have any other questions, please let me know.