There have been hints at Google working on their own VR hardware and software for a native integration with Android for quite some time, and Google officially confirmed all of the rumors and speculation during Google I/O back in May when they announced Daydream. While a large majority of the virtual world is being applied to games, apps in VR are not completely absent, and Google looks to now be setting up Chrome to work in VR so you can browse the web in a virtual space. Although not available in any stable capacity, Google already has builds of Chrome for Android in the developer and beta channels that contain open source WebVR standard support.
Surfing the internet would of course require a website to support virtual reality through the WebVR standard, but Google is helping to change this with something called a VR Shell that would allow users to freely browse any webpage without having to take their VR headset off, thus enabling the ability to move back and forth between VR-ready sites and those which have not yet added support for virtual reality. Having said that, Google's VR Shell is not yet finished and does not function completely, and there is no indication just yet that it will support anything outside of Google's Daydream platform and Google Cardboard headsets.
At the moment, the beta version of Chrome for Android does not yet include the VR Shell option, and is still limited to the Dev Channel build of the application, which means that a stable build of Chrome for Android that supports this feature could be a ways off. Presumably, Google might try to time the release of a public and stable build of Chrome for Android with these supported features around the same time as the first Daydream hardware, which is supposed to be launching sometime later on this year, although it's unclear as to what Google's plans are for updating Chrome for Android stable with support for VR. When Google does launch this version of Chrome for Android though, users should be able to browse the whole of the web in 360-degrees.