Lantern would only broadcast a specific subset of websites - Its essentially UUCP reinvented, so the idea you have to check the sizes likely won't work, or be necessary. Its one way so you shouldn't have to worry about the size of the data> its a broadcast medium like old fashioned TV or radio, rather than a two way medium like the internet. So, the main reason you're asking about this... well isn't going to be an issue. Wikipedia sums up outernet pretty well
Its an interesting problem though, so I'll give it a shot
The only way I can think of, that might work with some webpages is to use wget (with --spider so you don't download the page and --server-response to get the reported filesize). This approach has been heavily inspired by this SO question
This works with say superuser.com
[geek@phoebe os store]$ wget -v4 --spider --server-response superuser.com
Spider mode enabled. Check if remote file exists.
--2014-11-28 17:26:35-- http://superuser.com/
Resolving superuser.com (superuser.com)... 198.252.206.16
Connecting to superuser.com (superuser.com)|198.252.206.16|:80... connected.
HTTP request sent, awaiting response...
HTTP/1.1 200 OK
Cache-Control: public, no-cache="Set-Cookie", max-age=60
Content-Length: 71913
Content-Type: text/html; charset=utf-8
Expires: Fri, 28 Nov 2014 09:27:35 GMT
Last-Modified: Fri, 28 Nov 2014 09:26:35 GMT
Vary: *
X-Frame-Options: SAMEORIGIN
Set-Cookie: prov=85f6f157-7e84-43bf-b762-003cf7d8ff71; domain=.superuser.com; expires=Fri, 01-Jan-2055 00:00:00 GMT; path=/; HttpOnly
Date: Fri, 28 Nov 2014 09:26:34 GMT
Length: 71913 (70K) [text/html]
Remote file exists and could contain further links,
but recursion is disabled -- not retrieving.
[geek@phoebe os store]$ wget -v4 --spider --server-response http://superuser.com/questions/845893/is-it-possible-to-determine-through-the-internet-or-google-how-large-a-website/845895#845895
Spider mode enabled. Check if remote file exists.
--2014-11-28 17:26:43-- http://superuser.com/questions/845893/is-it-possible-to-determine-through-the-internet-or-google-how-large-a-website/845895
Resolving superuser.com (superuser.com)... 198.252.206.16
Connecting to superuser.com (superuser.com)|198.252.206.16|:80... connected.
HTTP request sent, awaiting response...
HTTP/1.1 200 OK
Cache-Control: public, no-cache="Set-Cookie", max-age=60
Content-Length: 69163
Content-Type: text/html; charset=utf-8
Expires: Fri, 28 Nov 2014 09:27:43 GMT
Last-Modified: Fri, 28 Nov 2014 09:26:43 GMT
Vary: *
X-Frame-Options: SAMEORIGIN
Set-Cookie: prov=7d270174-a377-4758-bbff-f4c87054de67; domain=.superuser.com; expires=Fri, 01-Jan-2055 00:00:00 GMT; path=/; HttpOnly
Date: Fri, 28 Nov 2014 09:26:42 GMT
Length: 69163 (68K) [text/html]
Remote file exists and could contain further links,
but recursion is disabled -- not retrieving.
But not, say
Google.com
[geek@phoebe os store]$ wget -v4 --spider --server-response google.com
Spider mode enabled. Check if remote file exists.
--2014-11-28 17:29:06-- http://google.com/
Resolving google.com (google.com)... 74.125.68.113, 74.125.68.138, 74.125.68.100, ...
Connecting to google.com (google.com)|74.125.68.113|:80... connected.
HTTP request sent, awaiting response...
HTTP/1.1 302 Found
Cache-Control: private
Content-Type: text/html; charset=UTF-8
Location: http://www.google.com.sg/?gfe_rd=cr&ei=YkB4VMT6F9iDoAO2tIH4Dw
Content-Length: 262
Date: Fri, 28 Nov 2014 09:29:06 GMT
Server: GFE/2.0
Alternate-Protocol: 80:quic,p=0.02
Location: http://www.google.com.sg/?gfe_rd=cr&ei=YkB4VMT6F9iDoAO2tIH4Dw [following]
Spider mode enabled. Check if remote file exists.
--2014-11-28 17:29:06-- http://www.google.com.sg/?gfe_rd=cr&ei=YkB4VMT6F9iDoAO2tIH4Dw
Resolving www.google.com.sg (www.google.com.sg)... 74.125.68.94
Connecting to www.google.com.sg (www.google.com.sg)|74.125.68.94|:80... connected.
HTTP request sent, awaiting response...
HTTP/1.1 200 OK
Date: Fri, 28 Nov 2014 09:29:06 GMT
Expires: -1
Cache-Control: private, max-age=0
Content-Type: text/html; charset=ISO-8859-1
Set-Cookie: PREF=ID=a1dfee7d97d41db1:FF=0:TM=1417166946:LM=1417166946:S=Uzy6MmaLU-UegGZU; expires=Sun, 27-Nov-2016 09:29:06 GMT; path=/; domain=.google.com.sg
Set-Cookie: NID=67=C_dkB1z4qdwwPkNMS80Ek1km-G4y716Evvh2BCEjYpdkpIJSAfXpjpTnSF496UlahPirO0Go-VhVxQjHlsEI_Hf4AxB9IfTyrGFzduyMB4rdTI-nX-kh0hlKhKQCrFg7; expires=Sat, 30-May-2015 09:29:06 GMT; path=/; domain=.google.com.sg; HttpOnly
P3P: CP="This is not a P3P policy! See http://www.google.com/support/accounts/bin/answer.py?hl=en&answer=151657 for more info."
Server: gws
X-XSS-Protection: 1; mode=block
X-Frame-Options: SAMEORIGIN
Alternate-Protocol: 80:quic,p=0.02
Transfer-Encoding: chunked
Length: unspecified [text/html]
Remote file exists and could contain further links,
but recursion is disabled -- not retrieving.