Hey @VirginTechnician! You nearly got it right formatting your code: you want to use triple backticks (on a US keyboard, the backtick is the key to the left of the 1 key and above the TAB key), rather than three apostrophes. But you have the right idea 
When you say a 'grouped' barplot, what did you have in mind? Did you mean having 'clusters' of bars, like the plot in this SO answer (or the clustered column chart in Excel)?
I can't help you plot this in base R (which is the system used by barplot(), but I can help you do it with ggplot2. The first thing you'll need to do is tidy your data. Right now, you have two variables (country and gender), leading to six categories. Instead of having a column for each 'series', or category, (Male/US, Female/US, Male/Canada, …), you'll want to have each column describe the value for each variable. So your data could look like this:
library(tidyverse)
culture <-
culture %>%
as_data_frame(rownames = NA) %>%
rownames_to_column(var = 'gender') %>%
gather(key = 'country', value = 'value', -gender)
#> # A tibble: 6 x 3
#> gender country value
#> <chr> <chr> <dbl>
#> 1 Female cananda 7.7
#> 2 male cananda 6.39
#> 3 Female United States 7.36
#> 4 male United States 6.43
#> 5 Female France 6.38
#> 6 male France 5.69
Now you can plot it with ggplot2 
ggplot(culture) +
geom_col(
aes(x = country, y = value, fill = gender),
position = 'dodge')
Created on 2018-10-25 by the reprex package (v0.2.0).
(You can swap country and gender depending on how you want to group it!)