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.