The myth that the Zionist owned the land of Palestine is explained below:

The first Zionist colony in Palestine, Petah Tiqva, was founded in 1878 on 14,200 dunums (dunum=1000 square meters) of land purchased from money lenders from Jaffa.  In 1901 the Karen Kayemeth (Jewish National Fund) was established in London to acquire land in Palestine.  Between the 1880s and 1914 some thirty Zionist colonies were founded and by 1914 the total Jewish population in Palestine had reached about eighty thousand.

United Nations map no. 94(b), of August 1950 shows the percentage of ownership of land in every subdistrict in Palestine under British mandate according to the village statistics of the British mandate government of Palestine.  The Map is reproduced in Walid Khalidi's book "Before their Diaspora", published 1984 by the Institute for Palestine Studies, Washington, DC Here are the numbers:
 

Subdistrict
Palestinian owned %
Zionist owned %
Public land %
Comments
Jerusalem 84 2 14
Hebron 96 < 1% 4
Beersheba 15 < 1% 85
Gaza 75 4 21
Ramleh 77 14 9
Ramallah 99 < 1% 1
Jaffa 47 39 14 Highest. Zionist TelAviv is here.
Nablus 87 < 1% 13
Tulkaram 78 17 5
Jenin 84 < 1% 16
Beisan 44 34 22
Nazereth 52 28 20
Haifa 42 35 23 
Tiberias 51 38 11
Acre 87 3 10
Safed 68 18 14

And by 1948 less than 7% of all the land in mandate Palestine was owned by the Zionists.