Josh, I have used also the function Minimize[] by fixing 6 or 7 arguments and minimizing the function f[] on the rest of variables and succeed to find the following local solution:

In[1]:= f[6.171439493048681`*^14, 2.3506806096409836`*^13, \
0.16717122698568856, 1.0555557671120848`, 2.9971, 1.3111, 0, 0]

Out[1]= 20784.9

I have fixed some variables and used Minimize[] for the rest of variables... Unfortunately, this is all I can do for the moment!

Apropos, Josh! You have done a mistake in writing argument of the function f[] in FindMinimum[]. The last but one argument must be dfr2 not drf2.

You may use the function FindMinimum[] to find a local solution of the sum of squares of the right parts of the equations:

In[110]:= Remove["Global`*"]

f[rin_, kmn_, wcr_, senc_, wfr_, senf_, dcr_, 
  dfr_] := (-33.323 + (
    4 rin)/((1 + 0.25^senc dcr^senc) (1 + 0.25^senf dfr^senf) (4 + 
       kmn)))^2 + (-83.446 + (
    7 rin)/((1 + 0.5^senc dcr^senc) (1 + 0.5^senf dfr^senf) (7 + 
       kmn)))^2 + (-498.46 + (
    16 rin)/((1 + 0.75^senc dcr^senc) (1 + 0.75^senf dfr^senf) (16 + 
       kmn)))^2 + (-1206.154 + (
    46 rin)/((1 + dcr^senc) (1 + dfr^senf) (46 + 
       kmn)))^2 + (-270.769 + (
    46 rin)/((46 + kmn) (1 + wcr^senc) (1 + wfr^
       senf)))^2 + (-36.923 + (
    4 rin)/((4 + kmn) (1 + 0.25^senc wcr^senc) (1 + 
       0.25^senf wfr^senf)))^2 + (-63.077 + (
    7 rin)/((7 + kmn) (1 + 0.5^senc wcr^senc) (1 + 
       0.5^senf wfr^senf)))^2 + (-147.962 + (
    16 rin)/((16 + kmn) (1 + 0.75^senc wcr^senc) (1 + 
       0.75^senf wfr^senf)))^2
FindMinimum[
 f[rin, kmn, wcr, senc, wfr, senf, dcr, 
  dfr], {{rin, 0}, {kmn, 0}, {wcr, 0}, {senc, 1}, {wfr, 0}, {senf, 
   1}, {dcr, 0}, {dfr, 0}}]

During evaluation of In[110]:= FindMinimum::nrlnum: The function value {58.7515 +1.24773*10^-6 I,64.166 +5.73835*10^-6 I,-302.257+0.0000286716 I,-71.1665+0.000314776 I,-<<19>>+<<22>> I,21.2778 +1.47581*10^-7 I,-11.9125+3.24322*10^-7 I,-126.799+5.43301*10^-7 I} is not a list of real numbers with dimensions {8} at {rin,kmn,wcr,senc,wfr,senf,dcr,dfr} = {43.8887,-0.325894,-0.164521,1.,-0.164521,1.,-0.804443,-0.804443}.

Out[112]= {103366., {rin -> 43.8887, kmn -> -0.325894, 
  wcr -> -0.164521, senc -> 1., wfr -> -0.164521, senf -> 1., 
  dcr -> -0.804443, dfr -> -0.804443}}

If the minimal value would be $0$, it meant the solution of the initial system of equations, As the found minimal values is not $0$, it means only that the function FindMinimum[] did not find the solution of the system of equations. We can only assume that the system doesn't have the solution.

Sure, you may select other initial point to search for other local solutions.

Perhaps there is no solution in the reals.

Could you use NMinimize and Norm or something similar and test this?