Switching to MMA10, I'd want to produce a plot using this code:
pad = 100; Fig8 = Plot[{2 p3[x], p2[x]}, {x, 0, 1}, Frame -> True, PlotStyle -> {{Thick, Black}, {Thick, Dashed, Gray}}, AspectRatio -> 0.6, PlotRange -> All, FrameLabel -> {{"Froeschlé HE", "Chirikov mLE"}, {"Chirikov mLE", "Froeschlé mLE"}}, PlotLegends -> Placed[LineLegend[{"Chirikov map", "2D Froeschlé map"}, LegendLabel -> Placed["p(x)", Above], LegendFunction -> "Frame", LegendMargins -> 1, LabelStyle -> 5], {0.65, 0.4}], BaseStyle -> {FontFamily -> "Times"}, FrameTicks -> {{{{0, "0.0"}, {0.2, "0.2"}, {0.4, "0.4"}, {0.6, "0.6"}, {0.8, "0.8"}, {1.0, "1.0"}}, {{0, "0.0"}, {0.2, "0.1"}, {0.4, "0.2"}, {0.6, "0.3"}, {0.8, "0.4"}, {1.0, "0.5"}}}, {{{0, "0.0"}, {0.2, "0.2"}, {0.4, "0.4"}, {0.6, "0.6"}, {0.8, "0.8"}, {1.0, "1.0"}}, {{0, "0.0"}, {0.2, "0.1"}, {0.4, "0.2"}, {0.6, "0.3"}, {0.8, "0.4"}, {1.0, "0.5"}}}}, ImagePadding -> {{pad, pad}, {All, All}}]` (Feel free to change the functions p2[x] and p3[x] to whatever you want)
This is generally the same approach I used since MMA7 and 'should' look nice. However, the output is unacceptable:
Export["Fig_8.eps",Fig8] 
compared to a .pdf file:
Export["Fig_8.pdf",Fig8]

Although I can convert the .pdf to .eps and it still looks ok, but this is not the case. Why does the .eps look so horrible in MMA10?
EDIT
To clarify: I don't want to get back to styling from older versions. I want the labels (axes and legend) not to be padded with randomly distributed spaces, e.g. bottom label should be "Chirikov mLE", not "Chi ri kovm LE" as it is. And the distances between letters in "Froeschle" seem to be too small. Compare the above screen shots from my Exported .eps and .pdf files.
Moreover, reading threads related to the one mentioned by Mr.Wizard, I have an impression that this lack of functionality comes from new underlying mechanism of Export in MMA10. The question is, can anyone figure out a workaround?
EDIT 2
I'm using Scientific Linux 6.

Export["Fig_8.eps", First@ImportString[ExportString[Fig8, "PDF"]]], or perhaps that is what you wish to avoid. You can make it into a function, so that it's not terribly inconvenient. I would say that it is also possible that certain OS don't do their fonts correctly. That's my experience as an end-user. $\endgroup$