Code with no warnings:

 $Version
  (* "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)" *)
 alpha = 1/2;
 R = 32;
 L = 45;
 a = 25;
 s = Table[(1 - Exp[-alpha])*
    NIntegrate[
     Exp[-alpha*Sqrt[(R^2 - 2*x*R*Cos[theta] + (x)^2) + z^2]], {z, 
      10*-a, 10*(L - a)}, {theta, 0, 2 Pi}, 
     Method -> "LocalAdaptive"], {x, 0, 32}(*,{a,0,L}*)]


  (*{5.70737*10^-6, 6.02507*10^-6, 7.03033*10^-6, 
   8.88889*10^-6, 0.0000119095, 0.000016599, 0.0000237522, \
  0.0000345936, 0.0000509949, 0.0000758066, 0.000113361, 0.000170229, \
  0.000256365, 0.000386811, 0.00058424, 0.00088272, 0.00133324, \
  0.00201175, 0.00303073, 0.00455555, 0.00682723, 0.0101931, 0.0151465, \
  0.0223752, 0.0328135, 0.0476838, 0.0684951, 0.0969276, 0.13446, \
  0.181468, 0.235262, 0.286114, 0.309488}*)

alpha = 0.04;(*Absorptionskoeffizient*)R = 32;(*Stabradius*)L = \
450;(*Stablänge*)a = 210;
x = 32;
s = NIntegrate[
Exp[-alpha*
 Sqrt[(R^2 - 2*x*R*Cos[theta] + (x)^2) + 
   z^2]]/(Sqrt[(R^2 - 2*x*R*Cos[theta] + (x)^2)]*
 Sqrt[1 + z^2/(R^2 - 2*x*R*Cos[theta] + (x)^2)]^3), {z, -a, 
L - a}, {theta, 0, 2 Pi}, Method -> "GaussKronrodRule"]

(*2.66833*)

Works fine.

alpha = 9/10;(*Absorptionskoeffizient*)R = 32;(*Stabradius*)
L = 45;(*Stablänge*)(*a=250;*);

f[x_?NumericQ, a_?NumericQ] := (1 - Exp[-alpha])*
  NIntegrate[
   Exp[-alpha*
      Sqrt[(R^2 - 2*x*R*Cos[theta] + (x)^2) + 
        z^2]]/(Sqrt[(R^2 - 2*x*R*Cos[theta] + (x)^2)]*
      Sqrt[1 + z^2/(R^2 - 2*x*R*Cos[theta] + (x)^2)]^3), {z, -10 a, 
    10*(L - a)}, {theta, 0, 2 Pi}, Method -> "LocalAdaptive", 
   MaxRecursion -> 100]

   f[31, 45](*x=31,a=45 ,OK*)
   (* 0.0184785 *)
   f[32, 45](*For x = 32 ,a=45 Not OK *)
   (* "The integrand  has evaluated to Overflow, Indeterminate, or Infinity  " *)
   f[33, 45](*x=33,a=45  OK*)
   (* 0.0179095*)

For certain values x,and a integral is divergent.

We can see:

 FunctionDomain[E^(-(9/10) Sqrt[1024 + x^2 + z^2 - 64 x Cos[theta]])/(
  Sqrt[1024 + x^2 - 
    64 x Cos[theta]] (1 + z^2/(1024 + x^2 - 64 x Cos[theta]))^(3/2)),
   x]
 (*x^2 - 64 x Cos[theta] > -1024 *)

 Table[x^2 - 64 x Cos[theta] > -1024 /. theta -> 2 Pi, {x, 31, 33}]
  (*{True, False, True}*)(* For x =32 is not true*)

Calculate this integral with NIntegrate?

Excellent catch. Thank you

I was scraping this before he put his code into a code block which made his multiplication more explicit. I think I correctly saw the switch in italics inside his integrand, but I missed the italic minus signs in his limits of integration.

Thank you again

This

alpha=1;R=2;x=3;a=4;L=5;
NIntegrate[Exp[-alpha*Sqrt[R^2-2*x*R*Cos[theta]+x^2+z^2]], {z,10-a,10(L-a)}, {theta,0,2 Pi}]

almost instantly gives

0.00743265

with no warnings or errors. So I assume your problem depends on the values of your constants.

Can you perhaps give the values of all your constants to show where and why there is a singularity? Possibly even include a 3d plot of the function to show where and how it blows up.

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