You can pivot your data frame into a longer format:
First add a column with the row number (bear number):
df = cbind("Bear"=as.factor(1:nrow(df)), df)
It needs to be factor so we can pass it as a group variable to ggplot. Now pivot:
df2 = tidyr::pivot_longer(df[,1:5], cols=2:5,
names_to="Year", values_to="Weight", names_prefix="Wt")
df2$Year = as.numeric(df2$Year)
We ignore the Length columns with df[,1:5]; say that we only want to pivot the weight columns with df[,2:5]; then say the name of the columns we want to create with names_to and values_to; and lastly the names_prefix="Wt" removes the "Wt" before the column names, leaving only the year number, but we get a character, so we need to make it numeric with as.numeric().
Then plot:
ggplot(df2, aes(x=Year, y=Weight, linetype=Bear)) + geom_line()
Output (Ps: i created my own data, so the actual numbers are off):

Just an addition, if you don't want to specify the columns of your dataset explicity, you can do:
df2 = df2[,grep("Wt|Bear", colnames(df)]
df2 = tidyr::pivot_longer(df2, cols=grep("Wt", colnames(df2)),
names_to="Year", values_to="Weight", names_prefix="Wt")
Edit: one plot for each group
You can use facet_wrap:
ggplot(df2, aes(x=Year, y=Weight, linetype=Bear)) +
facet_wrap(~Bear, nrow=2, ncol=4) +
geom_line()
Output:

You can change the nrow and ncol as you wish, and can remove the linetype from aes() as you already have a differenciation, but it's not mandatory.
You can also change the levels of the categorical data to make the labels on each graph better, do levels(df2$Bear) = paste("Bear", 1:7) for example (or do that the when creating it).
dput(your_data_frame)instead of the console.