You can change the text size of the quantities by passing a list defining the size to that argument like so:

plot(set1,  fills = list(fill = c("#009292", "#FFB6DB", "#B66DFF", "#6DB6FF"), alpha = 0.7),
                    labels = NULL, quantities = list(cex = .5), legend = list(labels = c("A", "B", "C", "D")))

This is one approach to adjusting the label and quantity sizes and providing a "margin".

Check out the documentation under plot.euler for other formatting options.

As eulerr is based on grid you can provide margins in effect by adjusting the size of the viewport used for plotting. The answer shows a plot reduced to 80% of the original height and width.

Although I noticed that the relative positions of the venn diagram elements changes as you rerun the code I never found that the circles straying outside the canvas margin.

set.seed() did not seem to change anything.

library(eulerr)
library(grid)

my_dt <- data.frame(
  value = c(1:8), 
  A = c(TRUE,  TRUE, FALSE, TRUE,  TRUE, FALSE, FALSE, TRUE),
  B = c(TRUE,  FALSE,  TRUE, TRUE, FALSE, FALSE, TRUE,  TRUE),
  C = c(FALSE, FALSE, FALSE, FALSE, TRUE,  TRUE,  TRUE, FALSE)
)

p <- plot(euler(my_dt[, c("A", "B", "C")]), 
     quantities = list(cex = 2),
     labels = list(cex = 2))

# edit the viewport dimensions

pwidth <- unit(0.8, "npc")
pheight <- unit(0.8, "npc")


p

^{Created on 2023-06-30 with reprex v2.0.2}

Sorry about this. That post is old and the arguments have changed.

Try this instead

fit2 <- euler(c(A = 16971, B = 218, C = 215, 
                "A&B" = 112, "A&C" = 112, "B&C"= 51,"A&B&C" = 23))
plot(fit2,
     fills = c("dodgerblue4", "darkgoldenrod1", "cornsilk4"),
     edges = FALSE,
     fontsize = 8,
     quantities = list(fontsize = 8))

When D. Knuth created his Computer Modern family, he used his Metafont for rendering the shapes. The sources for this rendering are "functions" (for each letters one function). These functions describe the shapes using Besier curves and metrics values are set too. The functions depend on many global parameters. There are 62 global parameters in Computer Modern family. A minor change of parameter values and rendering via the same functions results to the different font. For example \rm and \bf shapes are rendered with the same source functions but only different global parameters. The optical sizes are created with the same idea: only the global parameters are changed. It means that the different fonts from Computer Modern family (there are 75 fonts in it) were created by tuning the global Metafont parameters, no using only linear transformations.

There is text italic font in Computer Modern family: cmti10 and math italic font cmmi10. They are different fonts (they were rendered with different Metafont parameters) and they are intended for different purposes. The cmmi10 font is used in math mode for denoting variables and other math stuff, the cmti10 font is used for texts in italics. You show the cmmi10 in your first line (we must to read your line as f times u times n times c etc. because they are variables f, u, n, c and if they are one next to other then it means a product of variables in math formula). The cmti10 is shown in your second line (we read it as "functional typesetting"). You can see that not only spaces are different but also the shapes in your first line are slightly wider (due to the different Metafont parameters values).

The Euler font was created only for math formulas, no for text usage. So there is only math font zeurm10 and there isn't an alternative font for text usage. You can try to use it in text, but you must to copy the \fontdimen parameter used in text for spacing from another text font because these parameters are zero in math fonts:

\font\euler=zeurm10
\font\cmr=cmr10

This is normal text {\euler this is Euler without spacing parameters}.

\fontdimen2\euler=\fontdimen2\cmr % basic space
\fontdimen3\euler=\fontdimen3\cmr % stretching value
\fontdimen3\euler=\fontdimen4\cmr % shrinking value
\fontdimen7\euler=\fontdimen7\cmr % extra stretching value

And now: {\euler this is Euler with spacing parameters from cmr10}

\letterspacefont\eulersp\euler -30 % use it in pdfTeX only

\fontdimen2\eulersp=\fontdimen2\cmr % basic space
\fontdimen3\eulersp=\fontdimen3\cmr % stretching value
\fontdimen3\eulersp=\fontdimen4\cmr % shrinking value
\fontdimen7\eulersp=\fontdimen7\cmr % extra stretching value

And font: {\eulersp with smaller spaces between letters} 

\bye

The last example with \eulersp shows how you can set smaller spaces between letters, which is needed in text font. The \letterspacefont pdfTeX primitive was used here. If you are using another TeX engine then you have to use a different method for letter spacing.

Because the Euler font was intended for math typesetting, some important characters needed for text typesetting is missing here, for example comma, dot etc. So, if you want to use Euler fonts in text, you have to use a virtual font which combines the letters from various raw fonts.