See inline comments for changes.
The most significant change is using refline() to plot the regression lines based on the fits.
x1 = [20 40 60 80 100] ;
x2 = [20 40 60 80] ;
y1 = [0.240671275 0.552924282 0.874381642 0.99074338 1.354697701] ;
err1 = [0.052548669, 0.08814497, 0.117517615, 0.117123163, 0.167532477] ;
y2 = [0.11101623 0.11247439 0.208469577 0.322969453 0.386224488] ;
err2 = [0.133008916 0.125080979 0.1156091 0.11321253 0.119709187] ;
y3 = [0.111780925 0.162983696 0.31700554 0.378830489] ;
err3 = [0.056024757 0.050835675 0.071003682 0.058800409] ;
figure(1)
hold on % ADD THIS
xlim([0 105]) ;
scatter(x1,y1, 'g', 'Linewidth', 5)
Fit1 = polyfit(x1,y1,1);
% plot(polyval(Fit1,x1)); % REMOVE
rl(1) = refline(Fit1(1),Fit1(2)); % ADD THIS
rl(1).Color = 'g'; % ADD THIS
eb1 = errorbar(x1,y1,err1, 'vertical', 'LineStyle', 'none', 'Color', 'g', 'LineWidth', 2) ; % SET COLOR SAME AS LINE & LINE WIDTH
% set(eb1, 'color', 'k', 'LineWidth', 2); % NO LONGER NEEDED
% hold on % NO LONGER NEEDED, MOVED TOWARD THE TOP
scatter(x1,y2, 'r', 'Linewidth', 5)
Fit2 = polyfit(x1,y2,1);
% plot(polyval(Fit2,x1)); % REMOVE
rl(2) = refline(Fit2(1),Fit2(2)); % ADD THIS
rl(2).Color = 'r'; % ADD THIS
eb2 = errorbar(x1,y2,err2, 'vertical', 'LineStyle', 'none','color', 'r', 'LineWidth', 2) ; % SET COLOR SAME AS LINE & LINE WIDTH
% set(eb2, 'color', 'k', 'LineWidth', 2) % NO LONGER NEEDED
% hold on % NOT NEEDED, ONLY NEEDS TO BE RUN ONCE
scatter(x2,y3, 'b', 'Linewidth', 5)
Fit3 = polyfit(x2,y3,1);
% plot(polyval(Fit3,x2)); % REMOVE
rl(3) = refline(Fit3(1),Fit3(2)); % ADD THIS
rl(3).Color = 'b'; % ADD THIS
eb3 = errorbar(x2,y3,err3, 'vertical', 'LineStyle', 'none', 'color', 'b', 'LineWidth', 2) ;% SET COLOR SAME AS LINE & LINE WIDTH
% set(eb3, 'color', 'k', 'LineWidth', 2); % NOT NEEDED, ONLY NEEDS TO BE RUN ONCE
