This is a way

    In[8]:= With[{prC = #}, 
   FindMinimum[
    Sin[(x2 - x3)/2]^(-2) + Sin[(x4 - x3)/2]^(-2) + 
     Sin[(x4 - x2)/2]^(-2) + Sin[(x5 - x3)/2]^(-2) + 
     Sin[(x4 - x5)/2]^(-2) + Sin[(x5 - x2)/2]^(-2) + 
     Sin[(x2 + x3)/2]^(-2) + Sin[(x4 + x3)/2]^(-2) + 
     Sin[(x4 + x2)/2]^(-2) + Sin[(x5 + x3)/2]^(-2) + 
     Sin[(x4 + x5)/2]^(-2) + Sin[(x5 + x2)/2]^(-2) - 1, 
    SetPrecision[{{x2, -1.1}, {x3, 4.2}, {x4, 3.1}, {x5, 6.2}}, prC], 
    WorkingPrecision -> prC]
   ] & /@ {3, 32, 61}

    Out[8]= {{27.1, {x2 -> -1.06, x3 -> 4.25, x4 -> 3.15, x5 -> 6.29}}, 
   {27.000000000000000000000000000002, 
       {x2 -> -1.1071487177940905138496002853351, 
       x3 -> 4.2487413713838834891688579745400, 
       x4 -> 3.1415926535897932411309593731572, 
       x5 -> 6.2831853071795864767829818942239}}, \
   {27.00000000000000000000000000000000000000000000000000000000001, 
          {x2 -> -1.107148717794090503017065460178941989217603383476861620425671, 
           x3 -> 4.248741371383883741479708843458057352686502591415249448346964, 
           x4 -> 3.141592653589793238462643383279422163300186884667391231912264, 
           x5 -> 6.283185307179586476925286766559006139382728552267429532050186}}}

Thank you, this solution works indeed, even if I still don't understand why I get such a false result in my initial calculation.

But then I didn't find out how to set the precision for my initial values in such a way that I can modify it easily if I want to. I would like to write something like

4.2`n

which would mean

4.2`30

if n=30. But this doesn't seem to work.