It is possible to access and download the data via an API. The queries you can submit are also restricted, but it is possible to facilitate and automate the extraction by using some programming.
You can write a script to download the data in chunks and assemble them afterwards. Here is an example of how you could do it in R. A first chunk covers the years 2009-2011 and a second chunk the years 2012-2014. Here I have used copy-paste, but if you have more voluminous and more complex requests, you can use loops as well.
download.file("http://comtrade.un.org/api/get?fmt=csv&ps=2009,2010,2011&r=all&p=0&cc=TOTAL", "data1.csv", method = "auto", quiet=FALSE)
download.file("http://comtrade.un.org/api/get?fmt=csv&ps=2012,2013,2014&r=all&p=0&cc=TOTAL", "data2.csv", method = "auto", quiet=FALSE)
data1 <- read.table("data1.csv", header = T, sep = ",")
data2 <- read.table("data2.csv", header = T, sep = ",")
allthedata <- rbind(data1, data2)