This video goes through the process of signing up for a web hosting service with A2 hosting. You can literally have an instance of PostGIS up and running in half an hour for under $5/month that can be accessed from your own web map or from other clients such as QGIS.
What makes A2 Hosting so good for GIS applications is that they offer PostgreSQL databases in their standard cPanel package which means that you can easily install and use PostGIS. Most web hosting services only offer MySQL although you may be able to have them install PostgreSQL for you.
And even though I made a bit of a mistake in the video because I was using a domain that I already had registered, I would like to report that A2 hosting fixed that for me in about 2 hours after submitting a support ticket. I give them 5 stars for customer service on this occasion.
If you want to learn more about PostGIS please check out my course Introduction to spatial databases with PostGIS and QGIS, with over 11 hours of content, available now at Udemy.com for $20. And if you are interested in my other courses on Web GIS and QGIS please check out my course page on this blog.