Is it possible to store str output in a R object?

You can use capture.output:

l = capture.output(str(mtcars))
l
 [1] "'data.frame':\t32 obs. of  11 variables:"                         
 [2] " $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ..."  
 [3] " $ cyl : num  6 6 4 6 8 6 8 4 4 6 ..."                            
 [4] " $ disp: num  160 160 108 258 360 ..."                            
 [5] " $ hp  : num  110 110 93 110 175 105 245 62 95 123 ..."           
 [6] " $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ..."
 [7] " $ wt  : num  2.62 2.88 2.32 3.21 3.44 ..."                       
 [8] " $ qsec: num  16.5 17 18.6 19.4 17 ..."                           
 [9] " $ vs  : num  0 0 1 1 0 1 0 1 1 1 ..."                            
[10] " $ am  : num  1 1 1 0 0 0 0 0 0 0 ..."                            
[11] " $ gear: num  4 4 4 3 3 3 3 4 4 4 ..."                            
[12] " $ carb: num  4 4 1 1 2 1 4 2 2 4 ..."    

Building on thelatemail's approach slightly, it seems summary.default gives 3 bits of info for each column... so we can use a quick spread to have a slightly cleaner output:

iris %>% summary.default %>% as.data.frame %>% 
  dplyr::group_by(Var1) %>%  tidyr::spread(key = Var2, value = Freq)

# A tibble: 5 x 4
# Groups:   Var1 [5]
  Var1         Length Class  Mode   
  <fct>        <fct>  <fct>  <fct>  
1 Sepal.Length 150    -none- numeric
2 Sepal.Width  150    -none- numeric
3 Petal.Length 150    -none- numeric
4 Petal.Width  150    -none- numeric
5 Species      150    factor numeric

Not sure what Mode is but at least it's tidy.