If you look at length(Month) in your original code, you can see that it is empty.
Try this:
library(reshape)
library(tidyverse)
months <- c("Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec")
Month <- months
P1941TO1970 <- c(67.8, 53.1, 56.1, 81.5, 64.2, 56.9, 71.8, 71.1, 73.4, 67.5, 72.4, 60)
P1970TO2010 <- c(64, 57.8, 68.4, 79.1, 79.4, 84.9, 100.7, 79.2, 81.9, 77.4, 84.3, 73)
df <- data.frame(Month, P1941TO1970, P1970TO2010) %>%
mutate(Months = factor(months, level = months)) # factor here