Citizens of most countries need a medical health insurance for Russia valid for the duration of their trip in order to be able to get a Russian visa.
You might already have this insurance on your credit card policy, so it's worth checking that with your credit card issuer. They will need to send you a letter (per post or e-mail), which explicitly states that your travel insurance covers Russia for the duration of your trip (e.g. Russian visa validity period).
Also, if you're from UK, your NHS health insurance is valid in Russia and you can be treated at state hospitals for free (source), but it won't cover repatriation costs, non-health related accidents, and private treatments.
Citizens of all other countries, including USA, Canada, EU states, Australia, New Zealand, and Japan will need to buy a travel insurance that has medical and health coverage valid for the duration of their visa validity. This insurance can be bought through any insurance agency directly or online and usually costs about €30-€50 per week.
UK citizens are recommend to use Columbus Direct. Their plans start at about €40 per week and basically have some of the lowest prices available. You can apply online and your travel insurance document will arrive straight to your mailbox in a few minutes. You can then print it out and bring it, along with the other documents needed, to apply for the Russian visa.
Get Your Travel Insurance Online (UK)
Travel & Health Insurance for US and EU Citizens
USA, EU, Australian, Canadian and New Zealand citizens are recommended to use World Nomads. They have travel and health insurance plans that cover you worldwide, including flight delays and luggage loss, offered by the biggest insurers around the world.
Travel Insurance in Russia for German Citizens
Citizens of Germany can use HanseMerkur reisekrankenversicherung. The thing with Germany is that most insurers take ages to send you the documents and you don't want to be waiting for several days to get your insurance. HanseMerkur offers the full package online for only about €10 and it's instant and valid for the whole year. This insurance is also accepted by the Russian consulate (they have a list of insurance providers they accept on the consulate's website).
Ihre Reiseversicherung - Online-Buchung (Deutschland)
Travel Insurance for Students
If you're a student, you can try STA Travel as they offer lower prices to people under 25 and also World Nomads as they offer lower prices to people under 30.