What is a term for a server that is functioning 24/7?

Solution 1:

Critical services have their own terminology for this that lessens the ability to weasel around phrases like "24/7", "round-the-clock", "Mission critical", etc.

It's the "Nines" system and is common in legal contracts for "High availability" services.

Calculate "Nines" by taking the percentage of uptime over total time. For example, if a server was up for only 9 out of 10 seconds, that's 90% or "One nine".

"Five nines":

The common "Five nines" is 99.999% which still allows 6 seconds of downtime per week.

To get to less than 1 second per week, you need to guarantee "Six nines" or more -- which is a herculean task in the long term. (Even more-so if you don't use the standard "Unplanned" weasel word.)
For example, Google and Microsoft typically only promise "3 nines" (and frequently fail even that), which is a whopping 10 minutes down per week.

Solution 2:

round-the-clock

lasting all day and all night.
Examples:

• ‘round-the-clock surveillance’
• ‘He has hired 500 more workers and this month is adding a third shift for
  round-the-clock production - a first in auto manufacturing history.’

Oxford Dictionaries

Solution 3:

If the server cannot be down for even one second, the downtime it experiences must be zero seconds.

In other words, it is a zero-downtime server.

The term zero downtime seems to be widely used in the industry, for example here and here.

Solution 4:

I would go with high-availability.

Most, if not all servers, need some down-time when updates and patches are installed.

Solution 5:

"what is this particular type of server called in English"

I would go with either continuously available or mission critical. These terms better suit the specific context of a system or set of systems that support or underpin a goal or task. Personally I think the first specifically describes the type of system you are asking about.