What is the barrier in building your own interface with the payment provider? Is it red tape, or technical challenges?

Both. You have to build a pretty significant architecture which meets PCI compliance (so e.g. hardware firewalls, hosted in a data centre with sufficient physical security, etc. etc), which makes it hugely expensive, and technologically demanding. Then you have to sign a contract with a bank or banks in order to get the process going, which is the red tape part. Then you get to handle all the systems responsible for handling fraud, chargebacks, extra customer service, and so on. And then you pay higher rates because you're doing worthless volumes, even assuming that the banks will deal with you at those volumes.

Once you're done that, you've built a small-scale, inefficient, featureless, and altogether shitty version of what Stripe will offer up to anyone with an e-mail address, and it's taken you six to eight months to build. Oh and your app is out of date and all your competitors have taken your customers.

What I meant to ask was, "what did people do before Stripe came into existence"?

There are dozens of payment gateways. Braintree and Authorized.net are two large ones. If you accept credit card payments you will need a payment processor. You don't want to do this yourself.