|(Source: Alexa as of 29th September 2017)|
You can easily signup for a vk.com account, all you need is a valid mobile phone number. However, at times the system doesn't accept phone numbers for some unknown reason. In such instances you may seek the help of a generous friend who might want to help you use his/her account :)
Once you are logged in you can install a video downloader plugin such as Flash Video Downloader. As soon as the video starts playing, the plugin will capture the link to the video file and await your instruction to download the file.
Also, you may use Network Monitor, which is one of the Developer Tools provided by Firefox, to get the link of the video file. You simply have to play the video and wait for the link to a file of type mp4 to show up in your Network Monitor. Then you make use of that link to download the file.
You may still be able to play/download some videos without signing in although vk.com says so. This happens mostly when you are trying to play a video from the videos section. Instead, go to the profile page of the user who uploaded the video and you might be able to play the video and also download it using the above methods.