There are a variety of free or cheap options for serving geographical information online.
GoogleDocs allows you to add data, either privately or as a community effort, and display it as maps. You can find out more at MakeUseOf.
GitHub is probably the most flexible site for building free webpages that include some web-based mapping. GitHub offer free webspace through their GitHub Pages initiative. You can find more info on mapping on GitHub at Robin's blog. You can find out more about uploading to GitHub in the tutorial which is part of Advanced Programming. Many people use Jekyll to manage their GitHub pages, as it integrates automatically. You can find out more details on the Jekyll website and GitHub
Amazon offer Elastic Cloud Computing where you can essentially get hold of either a real or virtual computer of various specs, with adminsitrative control. Although there is a charging model you want to keep a strong eye on, for web servers there are options which are free with certain data/hit limits for 12 months. For more information about setting up in the cloud, see their Getting Started guide.