I have 3 solutions which could satisfy your requirements, those are: SeedDMS, JoomDOC and Microsoft SharePoint.
Both of them allow you to store document and manage its versions as well. You are able to access their repository via all popular variants of web browsers or via WebDAV protocol which enable users to browse the repository just like a folder in Windows Explorer like other traditional folders of Windows environment.
The aboves are their common aspects. Next, I'm describing things which make them differentiate with each other: 1/ Cost: SeedDMS is completely free while the others are not - in which SharePoint is the most expensive; 2/ Language: JoomDOC based on Joomla platform which is running PHP at its backend, SeedDMS is a standalone product also written by PHP while SharePoint is developed by ASP.NET; 3/ Windows Integration: SharePoint is the best integration and support with Windows environment (e.g. Active Directory, Access Controls, GPO, etc.) because it's developed by Microsoft; 4/ Community Support: SeedDMS is a new candidate so it has less support as compared with the others, the community of JoomDOC is also not good but we can get support from vendor while SharePoint is the best both for community and vendor as well.
In summary, if you mostly care about the cost or language used, SeedDMS and JoomDOC will be better for you. Otherwise, if functionalities and Windows support are most important, you could consider SharePoint as a best choice.