Line continuation in Ex commands is done by prepending a backslash to the beginning of the continuation lines themselves, not appending it to the end of the preceding lines.
See :help line-continuation:
Long lines in a :sourced Ex command script file can be split by inserting a line continuation symbol \ (backslash) at the start of the next line. There can be white space before the backslash, which is ignored.
And there's even an example for a :syntax command:
:syn match Comment
\ "very long regexp"
\ keepend
This may look very odd, so the help page even explains why it needs to be that way:
Rationale:
Most programs work with a trailing backslash to indicate line continuation. Using this in Vim would cause incompatibility with Vi.
For example for this Vi mapping:
:map xx asdf\
Therefore the unusual leading backslash is used.
Applying that to your particular case:
syntax region SomeName
\ start=/some_pattern/
\ end=/some other pattern/
\ contained
\ contains=SomeOtherName,YetAnotherName,
\EvenMoreNames,More,MoreAgain,
\KeepsGoing,Etc,
\UntilLineWrapsAroundAndItLooksNotThatBad
The help page also has detailed explanations on how to handle whitespace, which is applicable to the last lines in your example:
All leading whitespace characters in the line before a backslash are ignored. Note however that trailing whitespace in the line before it cannot be inserted freely; it depends on the position where a command is split up whether additional whitespace is allowed or not.
In short, you might need to be careful if using it where whitespace is to be prevented, but it's certainly doable even in those cases.