You'll have to do that by having two axes so that you can have two separate barh objects...I've railed at TMW for 20 years about how sorry their bar plots are in being so inflexible and difficult to work with.
That aside, you can begin with the venerable plotyy and then clean it up from there...
[hAx,hL,hR]=plotyy(bins5,5+counts5*0.5,bins50,50+counts50*0.5,@barh,@barh);
hAx(2).YLim=[5 8];
hAx(1).YLim=[5 8];
hAx(1).XLim=[1 125];
hAx(2).XLim=[1 125];
hAx(2).XTick=[];
hAx(2).YTick=[];
hAx(1).YTick=[5:8];
hAx(1).XScale='log';
hAx(2).XScale='log';
hAx(2).YLim=[5 8.5];
hAx(1).YLim=[5 8.5];
hAx(1).XTick=[1 5 50 100];
hR.BaseValue=50;
hL.BaseValue=5;
hR.ShowBaseLine='off';
hL.ShowBaseLine='off';
grid('on')
results in something close for starters...
And, you can be much more efficient in using set and handle arrays with array arguments; I just did this at command line on trial/error bais...
[hAx,hL,hR]=plotyy(bins5,c5,bins50,c50,@barh,@barh);
set(hAx,'YLim',[5 8.25]);
set(hAx,'XLim',[1 150]);
set(hAx,'XScale','log','XTick',[1 5 50 100],'YTick',[5:8]);
set(hAx(2),'XTick',[],'YTick',[]);
hR.BaseValue=50;
hL.BaseValue=5;
hB=[hL hR];
set(hB,'ShowBaseLine','off','BarWidth',1,'FaceColor','none');
set(hAx,'NextPlot','add')
hLn=plot(X,Y,'k-');
grid(hAx(1),'on')
