Plot and find intersection points of multiple curves

plot = Plot[{Sqrt[ν1 - ϵ], Sqrt[ν2 - ϵ], Sqrt[ν3 - ϵ], 
    Sqrt[ϵ] Tan[π/2 Sqrt[ϵ]], -Sqrt[ϵ] Cot[π/2 Sqrt[ϵ]]}, {ϵ, 0, 10}];

Graphics`Mesh`MeshInit[]

intersections = Graphics`Mesh`FindIntersections[plot];
Show[plot, Epilog -> {Red, PointSize[Large], Point@intersections}]

See also: Marking points of intersection between two curves

You can also do this non-graphically:

m = 9.1*10^-31;
L = .5;
hb = 1.055*10^-34;
e = hb^2/(2 m) (\[Pi]/L)^2 (1/(1.602*10^-19));
V1 = 2 e;
V2 = 5 e;
V3 = 8 e;
v1 = V1/e;
v2 = V2/e;
v3 = V3/e;
f1[x_] = Sqrt[v1 - x];
f2[x_] = Sqrt[v2 - x];
f3[x_] = Sqrt[v3 - x];
f4[x_] = Sqrt[x] Tan[\[Pi]/2 Sqrt[x]];
f5[x_] = -Sqrt[x] Cot[\[Pi]/2 Sqrt[x]];
functions = Plot[Evaluate@Through[{f1, f2, f3, f4, f5}[x]], {x, 0, 10}]

results = Quiet[Partition[
  Flatten[Thread[{#2, #1[#2]}] & @@@ Cases[{#1, 
     ToRules@(Reduce[{#1[x] == #2[x], 0 <= x <= 10}, x])} & @@@ 
       Subsets[{f1, f2, f3, f4, f5}, {2}], {a_, b__} -> {a, 
         ReplaceAll[x, {b}]}]], 2]]

(* {{0.463001, 1.23976}, {1.6379, 0.601749}, {0.600243, 
  2.09756}, {4.67897, 0.566599}, {2.31953, 1.63722}, {0.662803, 
  2.70873}, {5.60865, 1.5464}, {2.60346, 2.32305}}  *)

Show[functions, ListPlot[results, PlotStyle -> Black]]