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.

Note: If you put your Mathematica input on separate lines from your text and you put four spaces in front of each of those lines then the posting software will understand that this is Mathematica input and will do less desktop publishing, won't remove the * in your message, etc.

Your integrals seems in the beginning to be near zero, and NIntegrate seems to not like this. Look at

alpha = 0.5;
R = 32;
L = 45;
a = 25;
s = Table[{x, (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}]}, {x, 0, 32, 1}(*,{a,0,
   L}*)];
ListLinePlot[s, Epilog -> Point /@ s]

Looks reasonable - or?

@Bill Nelson: your integrand is not the same as in the problem.

Calculate this integral with NIntegrate?

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*)