The full symbolic solution may be just hopelessly complicated. I tried with numeric values for the parameters and the solution takes 2.2 MB of memory:
eqs = {\[CapitalLambda] - (Subscript[\[Alpha], 1] B +
Subscript[\[Beta], 1]
D + (Subscript[\[Alpha], 2] + Subscript[\[Beta],
2]) C + (Subscript[\[Alpha], 3] + Subscript[\[Beta],
3]) P) S - ((Subscript[\[Eta], 1] D +
Subscript[\[Eta], 2] C +
Subscript[\[Eta], 4] S) B + (Subscript[\[Eta], 3] C +
Subscript[\[Eta], 5] P) D) S - \[Mu] S ==
0, \[Lambda] + (Subscript[\[Alpha], 1] B +
Subscript[\[Alpha], 2] C +
Subscript[\[Alpha], 3] P) S - (Subscript[\[Rho], 1] D +
Subscript[\[Rho], 2] C + Subscript[\[Rho], 3] P + \[Mu] +
Subscript[\[Mu], 1] + Subscript[\[Epsilon], 1]) B ==
0, (Subscript[\[Beta], 1] D + Subscript[\[Beta], 2] C +
Subscript[\[Beta], 3] P) S - (\[Mu] + Subscript[\[Mu],
1]) D ==
0, (Subscript[\[Eta], 1]
D B + (Subscript[\[Eta], 2] B +
Subscript[\[Eta], 3] D) C + (Subscript[\[Eta], 4] B +
Subscript[\[Eta], 5] D) P) S - (\[Mu] + Subscript[\[Mu],
1]) C ==
0, (Subscript[\[Rho], 1] D + Subscript[\[Rho], 2] C +
Subscript[\[Rho], 3] P) B - (\[Mu] + Subscript[\[Mu], 1]) P ==
0, Subscript[\[Epsilon], 1] B - \[Mu] R == 0};
parameterValues = {Subscript[\[Alpha], 1] -> 1, \[Mu] -> 2,
Subscript[\[Beta], 1] -> -1, Subscript[\[Alpha], 2] -> 1,
Subscript[\[Beta], 2] -> -2, Subscript[\[Alpha], 3] -> 4,
Subscript[\[Beta], 3] -> 3, Subscript[\[Eta], 1] -> 1,
Subscript[\[Eta], 2] -> -1,
Subscript[\[Eta], 4] -> 1, \[Lambda] -> 1, \[CapitalLambda] -> 1/2,
Subscript[\[Eta], 3] -> 1, Subscript[\[Eta], 5] -> 2,
Subscript[\[Rho], 1] -> 1, Subscript[\[Rho], 2] -> 1,
Subscript[\[Rho], 3] -> 1, \[Mu] -> 1, Subscript[\[Mu], 1] -> -1,
Subscript[\[Epsilon], 1] -> 2};
Solve[eqs /. parameterValues, Reals]
